Inexact subgradient methods for semialgebraic functions

• 成果类型: Article; Early Access
• 署名作者: Bolte, Jerome; Le, Tam; Moulines, Eric; Pauwels, Edouard
• 署名单位: Universite de Toulouse; Universite Toulouse 1 Capitole; Toulouse School of Economics; Communaute Universite Grenoble Alpes; Institut National Polytechnique de Grenoble; Universite Grenoble Alpes (UGA); Centre National de la Recherche Scientifique (CNRS); Inria; Institut Polytechnique de Paris; Ecole Polytechnique; Universite de Toulouse; Universite Toulouse 1 Capitole; Toulouse School of Economics
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-025-02245-w
• 发表日期: 2025
• 关键词: stochastic approximations; CONVERGENCE; algorithms; REGULARITY; STABILITY

摘要

Motivated by the extensive application of approximate gradients in machine learning and optimization, we investigate inexact subgradient methods subject to persistent additive errors. Within a nonconvex semialgebraic framework, assuming boundedness or coercivity, we establish that the method yields iterates that eventually fluctuate near the critical set at a proximity characterized by an O(& varepsilon;rho)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\epsilon <^>\rho )$$\end{document} distance, where & varepsilon;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon $$\end{document} denotes the magnitude of subgradient evaluation errors, and rho\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} encapsulates geometric characteristics of the underlying problem. Our analysis comprehensively addresses both vanishing and constant step-size regimes. Notably, the latter regime inherently enlarges the fluctuation region, yet this enlargement remains on the order of & varepsilon;rho\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon <^>\rho $$\end{document}. In the convex scenario, employing a universal error bound applicable to coercive semialgebraic functions, we derive novel complexity results concerning averaged iterates. Additionally, our study produces auxiliary results of independent interest, including descent-type lemmas for nonsmooth nonconvex functions and an invariance principle governing the behavior of algorithmic sequences under small-step limits.